Coprime integers: $\gcd(a,b) = 1$

Integers $a$ and $b$ are coprime, or relatively prime, when

$\gcd(a,b) \;=\; 1$

(Common divisor, and the greatest common divisor $\gcd(a,b)$, with the convention $\gcd(0,0) := 0$). The relation is symmetric, since $\gcd(a,b) = \gcd(b,a)$ ($\gcd$ is symmetric and unchanged by signs: $\gcd(a,b) = \gcd(b,a) = \gcd(|a|,|b|)$; moreover $\gcd(a,0) = |a|$, $\gcd(a,1) = 1$, $\gcd(a,a) = |a|$, and $\gcd(a,b) \ge 1$ unless $a = b = 0$), and unchanged by signs, since $\gcd(a,b) = \gcd(|a|,|b|)$.

Boundary values. $\mathbb{N}$ contains $0$ and $\mathbb{Z}$ contains $0$ and $1$, so the three degenerate pairs are recorded explicitly, each read off $\gcd$ is symmetric and unchanged by signs: $\gcd(a,b) = \gcd(b,a) = \gcd(|a|,|b|)$; moreover $\gcd(a,0) = |a|$, $\gcd(a,1) = 1$, $\gcd(a,a) = |a|$, and $\gcd(a,b) \ge 1$ unless $a = b = 0$:

• $a$ and $1$ are coprime for every integer $a$, since $\gcd(a,1) = 1$; in particular $0$ and $1$ are coprime.
• $0$ and $b$ are coprime exactly when $b = 1$ or $b = -1$. Indeed $\gcd(0,b) = \gcd(b,0) = |b|$, so coprimality says $|b| = 1$, and by Absolute value in $\mathbb{Z}$: $|a| \ge 0$; $|a| = 0$ exactly when $a = 0$; $|-a| = |a|$; $|ab| = |a|\,|b|$; $-|a| \le a \le |a|$; and $|a| \le c$ exactly when $-c \le a \le c$ together with the case split of The absolute value $|a|$ of an integer this holds exactly for $b = 1$ and $b = -1$, the two units of $\mathbb{Z}$ ($(\mathbb{Z}, \cdot, 1)$ is a commutative monoid whose group of units is $\{1, -1\}$; equivalently $u \mid 1$ holds exactly for $u = 1$ and $u = -1$).
• $0$ and $0$ are not coprime, since $\gcd(0,0) = 0$ by the convention of Common divisor, and the greatest common divisor $\gcd(a,b)$, with the convention $\gcd(0,0) := 0$, and $0 \ne 1$: if $1$ were $0$ then $-1 = -0 = 0 = 1$, contradicting the two units being distinct ($(\mathbb{Z}, \cdot, 1)$ is a commutative monoid whose group of units is $\{1, -1\}$; equivalently $u \mid 1$ holds exactly for $u = 1$ and $u = -1$).

• Coprime is a statement about the pair, not about either integer. Neither $a$ nor $b$ need be prime, and primality is not defined anywhere on this page: $4$ and $9$ are coprime and neither is prime. What coprimality says is that the only common divisors are the units $1$ and $-1$ ($a$ and $b$ are coprime if and only if $ax + by = 1$ for some integers $x, y$; and in that case the only common divisors of $a$ and $b$ are $1$ and $-1$).

• Why it is worth a name before primes appear. The lemma that carries the weight in elementary number theory — if $a \mid bc$ and $a$ is coprime to $b$ then $a \mid c$ (If $\gcd(a,b) = 1$ and $a \mid bc$ then $a \mid c$; and if $a \mid c$, $b \mid c$ and $\gcd(a,b) = 1$ then $ab \mid c$) — needs coprimality and no primality at all. Primes enter later, and their key property is a special case of that lemma rather than an independent fact.